Definition:Conjugate Point/Dependent on N Functions

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Definition

Let $K$ be a functional such that:

$\displaystyle K \sqbrk h = \int_a^b \paren {\mathbf h'\mathbf P \mathbf h' + \mathbf h \mathbf Q \mathbf h} \rd x$

Consider Euler's equation related to the functional $K$:

$-\map {\dfrac \d {\d x} } {\mathbf P \mathbf h'} + \mathbf Q \mathbf h = 0$

where $\mathbf P$ and $\mathbf Q$ are symmetric matrices.

Let the general solution to this equation be:

$\set {\mathbf h^{\paren i} = \paren {\sequence {h_{ij} } }: i,j \in \N_{\le N} }$

Let:

$\exists j: \forall k \ne j: \paren {\map {\mathbf h^{\paren j} } a = 0} \land \paren {\map {h_{j j}'} a = 1, h'_{j k} = 0}$

Let the determinant, built from $h_{ij}$, be such that:

$\size {h_{i j} } \paren {\tilde a} = 0$

Here $i$ denotes rows, and $j$ denotes columns.


Then $\tilde a$ is said to be conjugate to point $a$ with respect to the functional $K$.


Sources

1963: I.M. Gelfand and S.V. Fomin: Calculus of Variations ... (previous) ... (next): $\S 5.29$: Generalization to n Unknown Functions